@Article{Gu2016,
author="Gu, Ran
and Huang, Fei
and Li, Xueliang",
title="Skew Randi'c matrix and skew Randi'c energy",
journal="Transactions on Combinatorics",
year="2016",
volume="5",
number="1",
pages="1-14",
abstract="‎Let $G$ be a simple graph with an orientation $\sigma$‎, ‎which‎ ‎assigns to each edge a direction so that $G^\sigma$ becomes a‎ ‎directed graph‎. ‎$G$ is said to be the underlying graph of the‎ ‎directed graph $G^\sigma$‎. ‎In this paper‎, ‎we define a weighted skew‎ ‎adjacency matrix with Rand\'c weight‎, ‎the skew Randi\'c matrix ${\bf‎ ‎R_S}(G^\sigma)$‎, ‎of $G^\sigma$ as the real skew symmetric matrix‎ ‎$[(r_s)_{ij}]$ where $(r_s)_{ij} = (d_id_j)^{-\frac{1}{2}}$ and‎ ‎$(r_s)_{ji} =‎ -‎(d_id_j)^{-\frac{1}{2}}$ if $v_i \rightarrow v_j$ is‎ ‎an arc of $G^\sigma$‎, ‎otherwise $(r_s)_{ij} = (r_s)_{ji} = 0$‎. ‎We‎ ‎derive some properties of the skew Randi\'c energy of an oriented‎ ‎graph‎. ‎Most properties are similar to those for the skew energy of‎ ‎oriented graphs‎. ‎But‎, ‎surprisingly‎, ‎the extremal oriented graphs‎ ‎with maximum or minimum skew Randi\'c energy are completely‎ ‎different‎, ‎no longer being some kinds of oriented regular graphs‎.",
issn="2251-8657",
doi="10.22108/toc.2016.9513",
url="http://toc.ui.ac.ir/article_9513.html"
}
@Article{Ramane2016,
author="Ramane, Harishchandra S.
and Nandeesh, K. Channegowda
and Gutman, Ivan
and Li, Xueliang",
title="Skew equienergetic digraphs",
journal="Transactions on Combinatorics",
year="2016",
volume="5",
number="1",
pages="15-23",
abstract="Let $D$ be a digraph with skew-adjacency matrix $S(D)$‎. ‎The skew‎ ‎energy of $D$ is defined as the sum of the norms of all‎ ‎eigenvalues of $S(D)$‎. ‎Two digraphs are said to be skew‎ ‎equienergetic if their skew energies are equal‎. ‎We establish an‎ ‎expression for the characteristic polynomial of the skew‎ ‎adjacency matrix of the join of two digraphs‎, ‎and for the‎ ‎respective skew energy‎, ‎and thereby construct non-cospectral‎, ‎skew equienergetic digraphs on $n$ vertices‎, ‎for all $n \geq 6$‎. ‎Thus we arrive at the solution of some open problems proposed in‎ ‎[X‎. ‎Li‎, ‎H‎. ‎Lian‎, ‎A survey on the skew energy of oriented graphs‎, ‎arXiv:1304.5707]‎. ",
issn="2251-8657",
doi="10.22108/toc.2016.9372",
url="http://toc.ui.ac.ir/article_9372.html"
}
@Article{Pattabiraman2016,
author="Pattabiraman, Kannan
and Kandan, P.",
title="Weighted Szeged indices of some graph operations",
journal="Transactions on Combinatorics",
year="2016",
volume="5",
number="1",
pages="25-35",
abstract="In this paper‎, ‎the weighted Szeged indices of Cartesian product and Corona product of two connected graphs are obtained‎. ‎Using the results obtained here‎, ‎the weighted Szeged indices of the hypercube of dimension $n$, Hamming graph‎, ‎$C_4$ nanotubes‎, ‎nanotorus‎, ‎grid‎, ‎$t-$fold bristled‎, ‎sunlet‎, ‎fan‎, ‎wheel‎, ‎bottleneck graphs and some classes of bridge graphs are computed‎.",
issn="2251-8657",
doi="10.22108/toc.2016.8594",
url="http://toc.ui.ac.ir/article_8594.html"
}
@Article{Acharya2016,
author="Acharya, Mukti
and Jain, Rashmi
and Kansal, Sangita",
title="ON $\bullet$-LICT signed graohs $L_{\bullet_c}(S)$ and $\bullet$-LINE signed graohs $L_\bullet(S)$",
journal="Transactions on Combinatorics",
year="2016",
volume="5",
number="1",
pages="37-48",
abstract="A signed graph (or‎, ‎in short‎, sigraph) $S=(S^u,\sigma)$ consists of an underlying graph $S^u‎ :‎=G=(V,E)$ and a function $\sigma:E(S^u)\longrightarrow \{+,-\}$‎, ‎called the signature of $S$‎. ‎A marking of $S$ is a function $\mu:V(S)\longrightarrow \{+,-\}$‎. ‎The canonical marking of a signed graph $S$‎, ‎denoted $\mu_\sigma$‎, ‎is given as $$\mu_\sigma(v)‎ :‎= \prod_{vw\in E(S)}\sigma(vw).$$‎ ‎The line graph of a graph $G$‎, ‎denoted $L(G)$‎, ‎is the graph in which edges of $G$ are represented as vertices‎, ‎two of these vertices are adjacent if the corresponding edges are adjacent in $G$‎. ‎There are three notions of a line signed graph of a signed graph $S=(S^u,\sigma)$ in the literature‎, ‎viz.‎, ‎$L(S)$‎, ‎$L_\times(S)$ and $L_\bullet(S)$‎, ‎all of which have $L(S^u)$ as their underlying graph; only the rule to assign signs to the edges of $L(S^u)$ differ‎. ‎Every edge $ee'$ in $L(S)$ is negative whenever both the adjacent edges $e$ and $e'$ in S are negative‎, ‎an edge $ee'$ in $L_\times(S)$ has the product $\sigma(e)\sigma(e')$ as its sign and an edge $ee'$ in $L_\bullet(S)$ has $\mu_\sigma(v)$ as its sign‎, ‎where $v\in V(S)$ is a common vertex of edges $e$ and $e'$‎. ‎‎The line-cut graph (or‎, ‎in short‎, lict graph) of a graph $G=(V,E)$‎, ‎denoted by $L_c(G)$‎, ‎is the graph with vertex set $E(G)\cup C(G)$‎, ‎where $C(G)$ is the set of cut-vertices of $G$‎, ‎in which two vertices are adjacent if and only if they correspond to adjacent edges of $G$ or one vertex corresponds to an edge $e$ of $G$ and the other vertex corresponds to a cut-vertex $c$ of $G$ such that $e$ is incident with $c$‎. ‎‎In this paper‎, ‎we introduce dot-lict signed graph (or $\bullet$-lict signed graph} $L_{\bullet_c}(S)$‎, ‎which has $L_c(S^u)$ as its underlying graph‎. ‎Every edge $uv$ in $L_{\bullet_c}(S)$ has the sign $\mu_\sigma(p)$‎, ‎if $u‎, ‎v \in E(S)$ and $p\in V(S)$ is a common vertex of these edges‎, ‎and it has the sign $\mu_\sigma(v)$‎, ‎if $u\in E(S)$ and $v\in C(S)$‎. ‎we characterize signed graphs on $K_p$‎, ‎$p\geq2$‎, ‎on cycle $C_n$ and on $K_{m,n}$ which are $\bullet$-lict signed graphs or $\bullet$-line signed graphs‎, ‎characterize signed graphs $S$ so that $L_{\bullet_c}(S)$ and $L_\bullet(S)$ are balanced‎. ‎We also establish the characterization of signed graphs $S$ for which $S\sim L_{\bullet_c}(S)$‎, ‎$S\sim L_\bullet(S)$‎, ‎$\eta(S)\sim L_{\bullet_c}(S)$ and $\eta(S)\sim L_\bullet(S)$‎, ‎here $\eta(S)$ is negation of $S$ and $\sim$ stands for switching equivalence‎.",
issn="2251-8657",
doi="10.22108/toc.2016.7890",
url="http://toc.ui.ac.ir/article_7890.html"
}
@Article{Eliasi2016,
author="Eliasi, Mehdi
and Ghalavand, Ali",
title="Ordering of trees by multiplicative second Zagreb index",
journal="Transactions on Combinatorics",
year="2016",
volume="5",
number="1",
pages="49-55",
abstract="‎For a graph $G$ with edge set $E(G)$‎, ‎the multiplicative second Zagreb index of $G$ is defined as‎ ‎$\Pi_2(G)=\Pi_{uv\in E(G)}[d_G(u)d_G(v)]$‎, ‎where $d_G(v)$ is the degree of vertex $v$ in $G$‎. ‎In this paper‎, ‎we identify the eighth class of trees‎, ‎with the first through eighth smallest multiplicative second Zagreb indeces among all trees of order $n\geq 14$‎. ",
issn="2251-8657",
doi="10.22108/toc.2016.9956",
url="http://toc.ui.ac.ir/article_9956.html"
}